Best birth control that doesn't involve weight gain?

Abstinence: First abstinence, then iud, then barrier protection (condom or dam). All the hormone based contraceptives involve an equivalent (minimal) risk of weight gain.

IUD/Implant: Regardless of your age, blood pressure, family history, or reactions to other birth control, the most effective and safest methods of birth control are long-acting progestin-only methods, such as the implant or the hormonal IUD, which decrease rates of endometrial cancer and decrease bleeding. They release low amounts of hormone daily and so will not affect your weight.